Interpretation of coefficient on interactive effects term in regression analysis

Question

I have two continuous variables: $X1$ and $X2$, both have a positive correlation on the dependent variable $y$ (continuous). I found that the interaction term $(X1*X2)$ is statistically significant for $p\lt(0.00)$ when added to the model. The final model is the following:

$$y=aX_1+bX_2-c(X_1*X_2)$$

Where $a$, $b$, and $c$ are the regression coefficients. However, I am wondering why the regression coefficient for $c$ of the interaction term is negative? What does it mean?

If I only add the interaction term $(X1*X2)$ to the model why is the coefficient positive when if I add it with $(X1)$ and $(X2)$ the regression coefficient becomes negative? It doesn't make sense!

Answer 1

If the following statement is true, Then it means y should increase as either X1 or X2 increases.

I have 2 continuous variables; X1 and X2, both have a positive correlation on the dependent variable y (continuous)

This is your final model:

$y=aX_1+bX_2-c(X_1*X_2)$

but I wonder why the regression coefficient c of the interaction term is negative? what does it mean?

you have a negative sign before the c coefficient which means it tries to decrease y as X1,X2 increases but the data necessitates that it should increase. Hence c is negative.

why if I add only the interaction term (X1*X2) to the model, the coefficient is positive while if I add it with (X1) and (X2) the regression coefficient becomes negative? it doesn't make sense!

By adding just X1*X2 to your does it look like the equation below?

$y=c(X_1*X_2)$

Then it makes sense only if c is positive if not then Y will decrease with increase in X1, X2. It makes sense.